Subgroups — #35223664

Order 20864.

394 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14579698	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₃₂₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#23781789	C₂	1	1	yes	maximal, characteristic, complemented
4	#23781743	C₂	1	1	yes	maximal, characteristic, complemented
5	#23781741	C₂	1	2	yes	maximal, complemented
6	#23781740	C₂	1	2	yes	maximal, complemented
7	#23781716	C₂	1	1	yes	maximal, characteristic, complemented
8	#23781715	C₂	1	2	yes	maximal, complemented
9	#23781688	C₂	1	2	yes	maximal, complemented
10	#23781687	C₂	1	1	yes	maximal, characteristic, complemented
11	#436×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579698	C₄	1	1	yes	characteristic
13	C₂×#14579683	C₂²	1	2	yes	—
14	C₂×#14579682	C₂²	1	2	yes	—
15	C₄×Dic₃₂₆	C₂²	1	1	yes	characteristic
16	#19104700	C₄	1	1	yes	characteristic
17	D₄×Dic₁₆₃	C₄	1	1	yes	characteristic
18	C₄×C₂²⋊D₁₆₃	C₂²	1	2	yes	—
19	(C₂²⋊D₁₆₃)⋊C₄	C₄	1	2	yes	—
20	#19104693	C₄	1	1	yes	characteristic
21	#19104691	C₄	1	2	yes	—
22	C₂×C₄×Dic₁₆₃	C₂²	1	2	yes	—
23	C₄²×D₁₆₃	C₂²	1	1	yes	characteristic
24	C₄⋊(C₁₆₃⋊C₈)	C₂²	1	1	yes	characteristic
25	Dic₁₆₃⋊C₈	C₂²	1	2	yes	—
26	Dic₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
27	Dic₁₆₃⋊C₈	C₂²	1	2	yes	—
28	Dic₁₆₃⋊C₈	C₂²	1	2	yes	—
29	Dic₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
30	C₂²⋊(C₁₆₃⋊C₈)	C₂²	1	2	yes	—
31	(C₂×D₁₆₃)⋊C₈	C₂²	1	1	yes	characteristic
32	(C₂×D₁₆₃)⋊C₈	C₂²	1	1	yes	characteristic
33	C₄×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
34	C₈×Dic₁₆₃	C₂²	1	2	yes	—
35	C₈×Dic₁₆₃	C₂²	1	2	yes	—
36	C₂×M₁₆×C₁₆₃	C₂²	1	2	yes	—
37	C₄×D₄×C₁₆₃	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₆₃	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₁₆₃	C₂²	1	2	yes	—
40	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
41	#14579698	—	2	4	no	—
42	C₂×Dic₃₂₆	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₁₆₃	C₂×C₄	1	2	yes	—
44	C₂²×Dic₁₆₃	C₂×C₄	1	2	yes	—
45	C₂²×Dic₁₆₃	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₁₆₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₆₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₆₃	C₂³	1	1	yes	characteristic
49	#14579683	—	4	2	no	—
50	#14579682	—	4	2	no	—
51	C₂×C₁₆₃⋊C₈	C₂³	1	1	yes	characteristic
52	C₂×C₁₆₃⋊C₈	C₂³	1	2	yes	—
53	C₂×C₁₆₃⋊C₈	C₂³	1	1	yes	characteristic
54	C₄⋊Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
55	Dic₁₆₃⋊C₄	C₂×C₄	1	2	yes	—
56	Dic₁₆₃⋊C₄	C₂×C₄	1	2	yes	—
57	C₂²⋊Dic₁₆₃	C₂×C₄	1	2	yes	—
58	(C₂×D₁₆₃)⋊C₄	C₂×C₄	1	2	yes	—
59	C₄×Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₁₆₃	C₂×C₄	1	2	yes	—
62	C₄×Dic₁₆₃	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₆₃	C₂³	1	2	yes	—
64	C₄×Dic₁₆₃	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₆₃	C₂³	1	2	yes	—
67	M₁₆×C₁₆₃	—	4	2	no	—
68	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₆₃	C₂³	1	2	yes	—
71	C₄⋊C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
72	C₂²⋊C₄×C₁₆₃	C₂×C₄	1	2	yes	—
73	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
74	C₂²×D₁₆₃	C₂²×C₄	1	1	yes	characteristic
75	Dic₃₂₆	M₁₆	1	4	yes	complemented
76	C₂²⋊D₁₆₃	—	2	4	no	—
77	C₂×Dic₁₆₃	—	2	4	no	—
78	C₂×Dic₁₆₃	—	2	4	no	—
79	C₂×Dic₁₆₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₁₆₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₆₃	C₂²×C₄	1	2	yes	—
82	C₂×Dic₁₆₃	C₂²×C₄	1	2	yes	—
83	C₂×Dic₁₆₃	C₂²×C₄	1	1	yes	characteristic
84	C₄×D₁₆₃	—	2	2	no	—
85	C₄×D₁₆₃	—	2	1	no	—
86	C₄×D₁₆₃	—	2	1	no	—
87	C₄×D₁₆₃	—	2	2	no	—
88	C₁₆₃⋊C₈	—	2	2	no	—
89	C₁₆₃⋊C₈	—	2	1	no	—
90	C₁₆₃⋊C₈	—	2	1	no	—
91	C₂³×C₁₆₃	C₂²×C₄	1	2	yes	—
92	D₄×C₁₆₃	—	2	2	no	—
93	C₂×C₄×C₁₆₃	—	2	2	no	—
94	C₂×C₄×C₁₆₃	—	2	2	no	—
95	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₆₃	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
99	C₈×C₁₆₃	—	2	1	no	—
100	C₈×C₁₆₃	—	2	2	no	—
101	C₈×C₁₆₃	—	2	1	no	—
102	C₂²×C₁₆₃	—	2	2	no	—
103	C₂²×C₁₆₃	—	2	2	no	—
104	C₂²×C₁₆₃	—	2	2	no	—
105	C₂²×C₁₆₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×D₁₆₃	—	2	1	no	—
107	C₂×D₁₆₃	—	2	1	no	—
108	C₂×D₁₆₃	—	2	1	no	—
109	C₄×C₁₆₃	—	2	2	no	—
110	C₄×C₁₆₃	—	2	1	no	—
111	C₄×C₁₆₃	C₂×M₁₆	1	2	yes	—
112	C₄×C₁₆₃	H(5,𝔽₂)	1	1	yes	characteristic
113	C₄×C₁₆₃	2₋¹⁺⁴	1	1	yes	characteristic
114	Dic₁₆₃	—	2	1	no	—
115	Dic₁₆₃	—	2	1	no	—
116	Dic₁₆₃	—	2	2	no	—
117	Dic₁₆₃	C₂×M₁₆	1	4	yes	—
118	Dic₁₆₃	#131	1	2	yes	—
119	C₂×C₁₆₃	—	4	2	no	—
120	C₂×C₁₆₃	#520	1	1	yes	characteristic
121	C₂×C₁₆₃	C₂²×M₁₆	1	1	yes	characteristic
122	C₂×C₁₆₃	C₂×#131	1	1	yes	characteristic
123	D₁₆₃	—	4	1	no	—
124	C₁₆₃	#2991	1	1	yes	characteristic, complemented, Sylow
125	#2991	—	163	1	no	maximal, Sylow
126	C₄×#41	—	163	1	no	—
127	C₈×Q₈	—	163	1	no	—
128	C₈×Q₈	—	163	1	no	—
129	#436	—	163	2	no	—
130	#436	—	163	1	no	—
131	#436	—	163	2	no	—
132	#436	D₁₆₃	1	1	yes	characteristic, complemented
133	#431	—	163	2	no	—
134	#431	—	163	1	no	—
135	#423	—	163	2	no	—
136	#423	—	163	1	no	—
137	C₂×#41	—	163	1	no	—
138	C₂×M₁₆	—	163	2	no	—
139	C₂×M₁₆	—	163	2	no	—
140	C₂×M₁₆	C₂×D₁₆₃	1	2	yes	—
141	C₄×Q₈	—	163	1	no	—
142	C₄×Q₈	—	163	1	no	—
143	C₄×D₄	—	163	1	no	—
144	C₄×D₄	—	163	2	no	—
145	C₄×D₄	—	163	2	no	—
146	C₄×D₄	C₂×D₁₆₃	1	1	yes	characteristic
147	#117	—	163	1	no	—
148	#117	—	163	2	no	—
149	C₂×C₄²	—	163	2	no	—
150	C₂×C₄²	—	163	1	no	—
151	C₄⋊C₈	—	163	2	no	—
152	C₄⋊C₈	—	163	1	no	—
153	C₄⋊C₈	—	163	2	no	—
154	C₄⋊C₈	—	163	2	no	—
155	C₄⋊C₈	—	163	1	no	—
156	C₄⋊C₈	—	163	1	no	—
157	C₄⋊C₈	C₂×D₁₆₃	1	1	yes	characteristic
158	C₂²⋊C₈	—	163	2	no	—
159	C₂²⋊C₈	—	163	1	no	—
160	C₂²⋊C₈	—	163	1	no	—
161	C₂²⋊C₈	C₂×D₁₆₃	1	2	yes	—
162	C₄×C₈	—	163	2	no	—
163	C₄×C₈	—	163	2	no	—
164	C₄×C₈	—	163	1	no	—
165	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
166	#41	—	326	4	no	—
167	C₂×Q₈	—	163	1	no	—
168	C₂×D₄	—	163	2	no	—
169	C₂×D₄	C₄×D₁₆₃	1	1	yes	characteristic
170	C₂²×C₄	—	163	2	no	—
171	C₂²×C₄	—	163	2	no	—
172	C₂²×C₄	—	163	1	no	—
173	C₂²×C₄	—	163	1	no	—
174	C₂²×C₄	—	163	1	no	—
175	C₂²×C₄	C₂²×D₁₆₃	1	2	yes	—
176	M₁₆	—	4	2	no	—
177	M₁₆	—	652	2	no	—
178	M₁₆	—	652	2	no	—
179	C₂×C₈	—	163	1	no	—
180	C₂×C₈	—	163	2	no	—
181	C₂×C₈	—	163	1	no	—
182	C₂×C₈	C₂²×D₁₆₃	1	1	yes	characteristic
183	C₂×C₈	C₂²×D₁₆₃	1	2	yes	—
184	C₂×C₈	C₂²×D₁₆₃	1	1	yes	characteristic
185	C₄⋊C₄	—	163	2	no	—
186	C₄⋊C₄	—	163	2	no	—
187	C₄⋊C₄	—	163	1	no	—
188	C₄⋊C₄	C₄×D₁₆₃	1	1	yes	characteristic
189	C₂²⋊C₄	—	163	2	no	—
190	C₂²⋊C₄	—	163	2	no	—
191	C₂²⋊C₄	C₄×D₁₆₃	1	2	yes	—
192	C₄²	—	163	2	no	—
193	C₄²	—	163	1	no	—
194	C₄²	—	163	1	no	—
195	C₄²	—	163	2	no	—
196	C₄²	—	163	2	no	—
197	C₄²	—	163	1	no	—
198	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
199	C₂³	—	163	1	no	—
200	C₂³	C₂×C₄×D₁₆₃	1	2	yes	—
201	Q₈	—	163	4	no	—
202	D₄	—	2	2	no	—
203	D₄	—	326	4	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	326	2	no	—
207	C₂×C₄	—	326	4	no	—
208	C₂×C₄	—	326	4	no	—
209	C₂×C₄	—	326	1	no	—
210	C₂×C₄	—	326	1	no	—
211	C₂×C₄	—	326	2	no	—
212	C₂×C₄	—	163	1	no	—
213	C₂×C₄	—	163	1	no	—
214	C₂×C₄	—	163	2	no	—
215	C₂×C₄	—	163	1	no	—
216	C₂×C₄	—	163	2	no	—
217	C₂×C₄	C₂×C₄×D₁₆₃	1	2	yes	—
218	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
219	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
220	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	2	2	no	—
223	C₈	—	2	1	no	—
224	C₈	—	326	2	no	—
225	C₈	—	326	1	no	—
226	C₈	—	326	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	326	1	no	—
231	C₂²	—	326	1	no	—
232	C₂²	—	326	1	no	—
233	C₂²	C₂²×C₄×D₁₆₃	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	1	no	—
236	C₄	—	326	1	no	—
237	C₄	—	326	1	no	—
238	C₄	—	326	2	no	—
239	C₄	—	163	4	no	—
240	C₄	—	163	2	no	—
241	C₄	M₁₆×D₁₆₃	1	2	yes	—
242	C₄	#19104780	1	1	yes	characteristic
243	C₄	#19104782	1	1	yes	characteristic
244	C₂	—	4	2	no	—
245	C₂	—	652	1	no	—
246	C₂	#23782144	1	1	yes	characteristic
247	C₂	C₂×M₁₆×D₁₆₃	1	1	yes	characteristic
248	C₂	#131×D₁₆₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₆₃
commutator, socle	C₂²×C₁₆₃	C₂³×C₄
Fitting	#436×C₁₆₃	C₂
prime radical	C₂×#14579698	C₄
cosocle	C₂×C₄×C₁₆₃	C₂⁴
hypercenter	#436	D₁₆₃
hypocenter	C₁₆₃	#2991
LCS	G ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₈×Dic₃₂₆	C₂	semidirect
3. #23781789	C₂	semidirect
4. #23781743	C₂	semidirect
5. #23781741	C₂	semidirect
6. #23781740	C₂	semidirect
7. #23781716	C₂	semidirect
8. #23781715	C₂	semidirect
9. #23781688	C₂	semidirect
10. #23781687	C₂	semidirect
11. #436×C₁₆₃	C₂	semidirect
75. Dic₃₂₆	M₁₆	semidirect
124. C₁₆₃	#2991	semidirect
132. #436	D₁₆₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×#14579698	2
C₈×Dic₃₂₆	2
#23781789	2
#23781743	2
#23781741	2
#23781740	2
#23781716	2
#23781715	2
#23781688	2
#23781687	2
#436×C₁₆₃	2
#2991	163